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The Fučík Spectrum: Exploring the Bridge Between Discrete and Continuous World

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Differential and Difference Equations with Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 47))

Abstract

In this paper, we would like to point out some similarities of interesting structures of Fučík spectra for continuous and discrete operators. We propose a simple algorithm that allows us to complete the reconstruction of the Fučík spectrum in the case of small order matrices. Finally, we point out some properties of the Fučík spectrum in general unifying terms and concepts.

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Notes

  1. 1.

    There are more than 300 qualitatively different patterns of the Fučík spectrum for 3 × 3 matrices (cf. the simple situation of 2 × 2 matrices in Fig. 4).

  2. 2.

    Let us note that examples in Sect. 2 are treated in H = L 2(0,π) ordered by \(K =\{ u \in H:\ u = u(x) \geq 0\ \ \mbox{ a.e. }x \in (0,\pi )\}\), and examples in Sect. 3 are treated in \(H = {\mathbb{R}}^{n}\) ordered by \(K = {(\mathbb{R}_{+})}^{n}\).

References

  1. Dancer, E.N.: On the Dirichlet problem for weakly non-linear elliptic partial differential equations. Proc. Roy. Soc. Edinb. Sect. A 76(4), 283–300 (1976/77)

    Google Scholar 

  2. Fučík, S.: Solvability of nonlinear equations and boundary value problems. Mathematics and its Applications, vol. 4. D. Reidel Publishing Co., Dordrecht (1980)

    Google Scholar 

  3. Holubová, G., Nečesal, P.: Nontrivial Fučík spectrum of one non-selfadjoint operator. Nonlinear Anal. 69(9), 2930–2941 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Holubová, G., Nečesal, P.: Nonlinear four-point problem: non-resonance with respect to the Fučík spectrum. Nonlinear Anal. 71(10), 4559–4567 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Holubová, G., Nečesal, P.: The Fučík spectra for multi-point boundary-value problems. Electron. J. Differ. Equat. Conf. 18, 33–44 (2010)

    Google Scholar 

  6. Holubová, G., Nečesal, P.: Fučík spectrum in general. To appear in Topol. Methods Nonlinear Anal. (2011)

    Google Scholar 

  7. Hyers, D.H., Isac, G., Rassias, T.M.: Topics in Nonlinear Analysis and Applications. World Scientific Publishing Co. Inc., River Edge (1997)

    Book  MATH  Google Scholar 

  8. Sergejeva, N.: On nonlinear spectra for some nonlocal boundary value problems. Math. Model. Anal. 13(1), 87–97 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Stehlík, P.: Discrete Fučík spectrum - anchoring rather than pasting. Boundary Value Problems, 2013:67 (2013)

    Article  Google Scholar 

  10. Švarc, R.: The solution of a Fučík’s conjecture. Commentat. Math. Univ. Carol. 25, 483–517 (1984)

    MATH  Google Scholar 

  11. Švarc, R.: Two examples of the operators with jumping nonlinearities. Commentat. Math. Univ. Carol. 30(3), 587–620 (1989)

    MATH  Google Scholar 

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Acknowledgement

The authors were supported by the Ministry of Education, Youth and Sports of the Czech Republic, Research Plan MSM4977751301, by grant ME09109 (program KONTAKT) and partially by the European Regional Development Fund (ERDF), project “NTIS—New Technologies for Information Society”, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.

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Authors and Affiliations

  1. University of West Bohemia, Univerzitní 22, 306 14, Plzeň, Czech Republic

    Gabriela Holubová & Petr Nečesal

Authors
  1. Gabriela Holubová
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  2. Petr Nečesal
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Corresponding author

Correspondence to Gabriela Holubová .

Editor information

Editors and Affiliations

  1. Departamento de Ciências Exactas e Naturais, Academia Militar, Amadora, Portugal

    Sandra Pinelas

  2. University of Zürich Institute of Mathematics, Zürich, Switzerland

    Michel Chipot

  3. Masaryk University Dept. Mathematics, Brno, Czech Republic

    Zuzana Dosla

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Holubová, G., Nečesal, P. (2013). The Fučík Spectrum: Exploring the Bridge Between Discrete and Continuous World. In: Pinelas, S., Chipot, M., Dosla, Z. (eds) Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics, vol 47. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7333-6_36

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